Let \(E\)  be an elliptic curve over \(\mathbb {Q}\)  with good ordinary reduction at \(p\)  for which the Galois module \(E[p]\)  is irreducible. Denote by \(\mathbb {Q}_{\infty }(E):=\mathbb {Q}(E_{p^{\infty }})\)  be the field obtained by adjoining the coordinates of the group of \(p\) -power division points of \(E\) . Let \(G_E:=\operatorname{Gal}(\mathbb {Q}_{\infty }(E)/\mathbb {Q})\)  and identify \(G_E\)  with its image in \(\operatorname{Aut}(T_p(E))\simeq \text{GL}_2(\mathbb {Z}_p)\) , where as usual \(T_p(E):=\varprojlim _n E_{p^n}\)  is the \(p\) -adic Tate-module of \(E\) . By a well-known theorem of Serre, \(G_E\)  is an open subgroup of \(\text{GL}_2(\mathbb {Z}_p)\) . Let \(H_E:=\operatorname{Gal}(\mathbb {Q}_{\infty }(E)/\mathbb {Q}^{cyc})\)  and as usual, \(\Gamma =G_E/H_E\) . It was shown by Kato [1]} that \(\operatorname{Sel}(E/\mathbb {Q}^{cyc})\)  is a cotorsion \(\Lambda (\Gamma )\) -module. It follows from this that the Selmer group \(\operatorname{Sel}(E/\mathbb {Q}_{\infty }(E))\)  is a cotorsion \(\Lambda (G_E)\) -module (cf. [2]}). Let \(\mathcal {M}_E\)  be the set of primes at which the \(j\) -invariant of \(E\)  is non-integral.
